[proofplan] This entry records the Polynomial Freiman-Ruzsa conjecture in the torsion-free integer setting, not a proved theorem. The statement has therefore been classified as a conjecture and includes the nonempty hypothesis on $A$ and the parameter condition $K \geq 1$. The proof field records the mathematical status and the degenerate $K = 1$ case, rather than presenting an invalid derivation of a currently open result. [/proofplan] [step:Record the conjectural assertion and the degenerate endpoint] The hypotheses are that $A \subset \mathbb{Z}$ is finite and nonempty and that there is a parameter $K \geq 1$ such that \begin{align*} |A + A| \leq K |A|. \end{align*} The asserted conjectural conclusion is the existence of an absolute constant $C > 0$, a proper generalized arithmetic progression $P \subset \mathbb{Z}$ of rank at most $C \log K$, and a finite set $X \subset \mathbb{Z}$ with $|X| \leq K^C$ such that $A \subset X + P$ and $|P| \leq K^C |A|$, in the standard sense of $K^C$-control. At the endpoint $K = 1$, the inequality gives $|A + A| \leq |A|$. Since $A$ is a finite nonempty subset of the torsion-free group $\mathbb{Z}$, the elementary sumset inequality $|A + A| \geq 2|A| - 1$ forces $|A| = 1$. In that case $A = \{a\}$ for some $a \in \mathbb{Z}$, and the rank-zero proper generalized arithmetic progression $P = \{a\}$ gives the required control with $X = \{0\}$. For $K > 1$, this conclusion is precisely the Polynomial Freiman-Ruzsa conjecture in the torsion-free integer setting. Therefore the current entry cannot be completed as a theorem proof by citing established results or by formalizing a known argument. A valid proof would constitute a resolution of the conjecture in this setting. [/step] [step:Conclude that the entry is a conjecture rather than a proved theorem] Because the requested conclusion is conjectural for $K > 1$, the proof field cannot establish the statement as a theorem. The mathematically sound correction is to classify the entry as a conjecture, while retaining the endpoint check above for $K = 1$. Under this conjectural statement, there is no publication-quality proof to provide for the open range $K > 1$. [/step]